Z normal table question

Question

I am trying to calculate $$P(Z > -3)$$

Using the table here http://www.utstat.toronto.edu/mikevans/jeffrosenthal/AppendixD.pdf

This only goes up to $-3.4$

It says that $P(Z < -1) = 1 - P(P < 1)$. But what about if its greater than?

Answer 1

The normal distribution is symmetric: $Z \overset{d}{\sim} -Z$. To say that $Z$ and $-Z$ have the same distribution means that $\mathbf{P}(Z \in A) = \mathbf{P}(-Z \in A)$ for any set $A$. For example if we take $A = \{z : z > a\}$ then $Z \in A$ if and only if $Z > a$, so we obtain

$$\mathbf{P}(Z > a) = \mathbf{P}(-Z > a) = \mathbf{P}(Z < -a). $$

We also know that if two events are complementary then their total probability is $1$. For example:

$$ \mathbf{P}(Z > a) + \mathbf{P}(Z \le a) = 1$$

Since $Z$ is either $> a$ or $\le a$ 100% of the time.

Applying these two rules, we get the following: $$ \mathbf{P}(Z > -3) = \mathbf{P}(Z < 3) = 1 - \mathbf{P}(Z \le -3) = 1 - \mathbf{P}(Z \ge 3). $$

Also $\mathbf{P}(Z \ge a) = \mathbf{P}(Z > a)$ and $\mathbf{P}(Z \le a) = \mathbf{P}(Z < a)$.